Moderate Deviations Analysis of Binary Hypothesis Testing

This paper is focused on the moderate-deviations analysis of binary hypothesis testing. The analysis relies on a concentration inequality for discrete-parameter martingales with bounded jumps, where this inequality forms a refinement to the Azuma-Hoeffding inequality. Relations of the analysis to the moderate deviations principle for i.i.d. random variables and to the relative entropy are considered.Comment: Presented at the 2012 IEEE International Symposium on Information Theory (ISIT 2012) at MIT, Boston, July 2012. It appears in the Proceedings of ISIT 2012 on pages 826-83

Moderate Deviation Analysis for Classical Communication over Quantum Channels

© 2017, Springer-Verlag GmbH Germany. We analyse families of codes for classical data transmission over quantum channels that have both a vanishing probability of error and a code rate approaching capacity as the code length increases. To characterise the fundamental tradeoff between decoding error, code rate and code length for such codes we introduce a quantum generalisation of the moderate deviation analysis proposed by Altŭg and Wagner as well as Polyanskiy and Verdú. We derive such a tradeoff for classical-quantum (as well as image-additive) channels in terms of the channel capacity and the channel dispersion, giving further evidence that the latter quantity characterises the necessary backoff from capacity when transmitting finite blocks of classical data. To derive these results we also study asymmetric binary quantum hypothesis testing in the moderate deviations regime. Due to the central importance of the latter task, we expect that our techniques will find further applications in the analysis of other quantum information processing tasks

Asymptotic Estimates in Information Theory with Non-Vanishing Error Probabilities

This monograph presents a unified treatment of single- and multi-user problems in Shannon's information theory where we depart from the requirement that the error probability decays asymptotically in the blocklength. Instead, the error probabilities for various problems are bounded above by a non-vanishing constant and the spotlight is shone on achievable coding rates as functions of the growing blocklengths. This represents the study of asymptotic estimates with non-vanishing error probabilities. In Part I, after reviewing the fundamentals of information theory, we discuss Strassen's seminal result for binary hypothesis testing where the type-I error probability is non-vanishing and the rate of decay of the type-II error probability with growing number of independent observations is characterized. In Part II, we use this basic hypothesis testing result to develop second- and sometimes, even third-order asymptotic expansions for point-to-point communication. Finally in Part III, we consider network information theory problems for which the second-order asymptotics are known. These problems include some classes of channels with random state, the multiple-encoder distributed lossless source coding (Slepian-Wolf) problem and special cases of the Gaussian interference and multiple-access channels. Finally, we discuss avenues for further research.Comment: Further comments welcom

Discrete Optimization for Interpretable Study Populations and Randomization Inference in an Observational Study of Severe Sepsis Mortality

Motivated by an observational study of the effect of hospital ward versus intensive care unit admission on severe sepsis mortality, we develop methods to address two common problems in observational studies: (1) when there is a lack of covariate overlap between the treated and control groups, how to define an interpretable study population wherein inference can be conducted without extrapolating with respect to important variables; and (2) how to use randomization inference to form confidence intervals for the average treatment effect with binary outcomes. Our solution to problem (1) incorporates existing suggestions in the literature while yielding a study population that is easily understood in terms of the covariates themselves, and can be solved using an efficient branch-and-bound algorithm. We address problem (2) by solving a linear integer program to utilize the worst case variance of the average treatment effect among values for unobserved potential outcomes that are compatible with the null hypothesis. Our analysis finds no evidence for a difference between the sixty day mortality rates if all individuals were admitted to the ICU and if all patients were admitted to the hospital ward among less severely ill patients and among patients with cryptic septic shock. We implement our methodology in R, providing scripts in the supplementary material

Decision trees in epidemiological research

Background: In many studies, it is of interest to identify population subgroups that are relatively homogeneous with respect to an outcome. The nature of these subgroups can provide insight into effect mechanisms and suggest targets for tailored interventions. However, identifying relevant subgroups can be challenging with standard statistical methods. Main text: We review the literature on decision trees, a family of techniques for partitioning the population, on the basis of covariates, into distinct subgroups who share similar values of an outcome variable. We compare two decision tree methods, the popular Classification and Regression tree (CART) technique and the newer Conditional Inference tree (CTree) technique, assessing their performance in a simulation study and using data from the Box Lunch Study, a randomized controlled trial of a portion size intervention. Both CART and CTree identify homogeneous population subgroups and offer improved prediction accuracy relative to regression-based approaches when subgroups are truly present in the data. An important distinction between CART and CTree is that the latter uses a formal statistical hypothesis testing framework in building decision trees, which simplifies the process of identifying and interpreting the final tree model. We also introduce a novel way to visualize the subgroups defined by decision trees. Our novel graphical visualization provides a more scientifically meaningful characterization of the subgroups identified by decision trees. Conclusions: Decision trees are a useful tool for identifying homogeneous subgroups defined by combinations of individual characteristics. While all decision tree techniques generate subgroups, we advocate the use of the newer CTree technique due to its simplicity and ease of interpretation